Characterization theorem for Laguerre-Hahn orthogonal polynomials on non-uniform lattices

TL;DR

This paper establishes a characterization theorem for Laguerre-Hahn orthogonal polynomials on non-uniform lattices, linking Riccati equations and difference relations to deepen understanding of their structure.

Contribution

It provides a new theorem that characterizes Laguerre-Hahn orthogonal polynomials on non-uniform lattices through equivalences involving Riccati equations and difference relations.

Findings

Proves the equivalence between Riccati equations and difference relations

Characterizes Laguerre-Hahn orthogonal polynomials on non-uniform lattices

Enhances understanding of polynomial structure and relations

Abstract

It is stated and proved a characterization theorem for Laguerre-Hahn orthogonal polynomials on non-uniform lattices. This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.